By the end of 4th grade, students should have learned these fundamentals of fractions:
1. A fraction is a part of a whole. When a whole is divided into equal-sized pieces, a fraction of the whole is one or more of those pieces. The numerator of the fractions tells how many pieces there are, and the denominator tells how many pieces the whole was divided into.
For example, a pizza cut into 8 slices can represent eighths. One slice is 1/8. 4 slices is 4/8. (It's hard to make proper fractions on wikispaces, so please don't be annoyed at the use of the diagonal fraction bar. You should always use the horizontal fraction bar with young students.)
2. Manipulatives and drawings are good ways of representing fractions. Circle fractions can be used as direct models of pizza slices (like the drawing to the left.) Other representations include bar models (think about 1/8 of a Tootsie Roll) and area models (1/8 of a pan of brownies cut 4x2).
3. Rulers are marked to show fractions of one inch. Number lines can be drawn to show that there are numbers (fractions) between integers.
4. Equivalent fractions are the same size (the same total amount of the whole) but they are divided into different numbers of pieces. For example, if you cut a pizza into fourths, one section would be 1/4. If you cut a pizza of the same size into eighths, two sections of that pizza would be 2/8. Both of these sections are the same size.
5. Fractions can also be used to represent parts of a set of object. For example, if there are 10 red M&M's in a bag of 50 M&M's, then the fraction of red M&M's is 10/50.
6. Equivalent fractions can be found by "scaling up" or "scaling down." This makes sense when you think of a fraction as part of a set of objects. In the M&M example above, if there is always the same fraction of red M&M's in every bag, then a bag of 25 (half the original bag) would have 5 red M&M's (half the original amount). The fraction is 5/25, which is equivalent to 10/50. In this case, we have scaled down by a factor of 2, or divided both the part and the whole by 2. In a bag of 100 M&M's, there would be 20 red M&M's, scaling up by a factor of 2 - multiplying both the part and the whole by 2.
7. "Mixed numbers" are numbers that have both an integer part and a fraction part, like 5 1/3. On the number line, this would be 1/3 of the way from 5 to 6. As a real object, this might be 5 1/3 cups of flour in a cake recipe (5 cups and 1/3 cup more). Mixed numbers can also be represented as "improper fractions," an equivalent number with no integer part. In improper fractions, the numerator is larger than the denominator. For example, 5 1/3 is equivalent to 16/3.
8. Decimal numbers can be written as fractions with denominators of 10, 100, 1000, etc. The first place to the right of the decimal represents tenths, the second decimal place represents hundredths, the third decimal place represents thousandths, etc. For example, the number 7.3 is equivalent to 7 3/10. The number 20.45 is equivalent to 20 45/100.
9. Not all fractions can be represented by terminating decimal numbers. For example, 1/3 has the decimal equivalent of 0.333333... a repeating, non-terminating decimal (it goes on forever).
10. Fractions are also a way of writing a division statement. 4/5 means 4 divided by 5. You can use this concept to find the decimal equivalent for a fraction: 4 divided by 5 = 0.8.
11. Percents represent parts out of 100. 50% means 50 parts out of 100. This is equivalent to the fraction 50/100, or the decimal 0.5.
Assignment: Use the first section of the Student Packet (Basic Fraction Concepts Assessment) to assess your students' understanding of basic fraction ideas. Since the handout is in Microsoft Word, you can add, delete or change questions as you see fit. Students who don't have a grasp of these basic ideas in 5th grade or above probably need additional support to learn them before going further.
Basic ideas about decimal numbers and percents (Decimals and Percents Assessment) are assessed in the last section of the Student Packet.
Fraction Basics
By the end of 4th grade, students should have learned these fundamentals of fractions:
1. A fraction is a part of a whole. When a whole is divided into equal-sized pieces, a fraction of the whole is one or more of those pieces. The numerator of the fractions tells how many pieces there are, and the denominator tells how many pieces the whole was divided into.
For example, a pizza cut into 8 slices can represent eighths. One slice is 1/8. 4 slices is 4/8. (It's hard to make proper fractions on wikispaces, so please don't be annoyed at the use of the diagonal fraction bar. You should always use the horizontal fraction bar with young students.)
3. Rulers are marked to show fractions of one inch. Number lines can be drawn to show that there are numbers (fractions) between integers.
4. Equivalent fractions are the same size (the same total amount of the whole) but they are divided into different numbers of pieces. For example, if you cut a pizza into fourths, one section would be 1/4. If you cut a pizza of the same size into eighths, two sections of that pizza would be 2/8. Both of these sections are the same size.
5. Fractions can also be used to represent parts of a set of object. For example, if there are 10 red M&M's in a bag of 50 M&M's, then the fraction of red M&M's is 10/50.
6. Equivalent fractions can be found by "scaling up" or "scaling down." This makes sense when you think of a fraction as part of a set of objects. In the M&M example above, if there is always the same fraction of red M&M's in every bag, then a bag of 25 (half the original bag) would have 5 red M&M's (half the original amount). The fraction is 5/25, which is equivalent to 10/50. In this case, we have scaled down by a factor of 2, or divided both the part and the whole by 2. In a bag of 100 M&M's, there would be 20 red M&M's, scaling up by a factor of 2 - multiplying both the part and the whole by 2.
7. "Mixed numbers" are numbers that have both an integer part and a fraction part, like 5 1/3. On the number line, this would be 1/3 of the way from 5 to 6. As a real object, this might be 5 1/3 cups of flour in a cake recipe (5 cups and 1/3 cup more). Mixed numbers can also be represented as "improper fractions," an equivalent number with no integer part. In improper fractions, the numerator is larger than the denominator. For example, 5 1/3 is equivalent to 16/3.
8. Decimal numbers can be written as fractions with denominators of 10, 100, 1000, etc. The first place to the right of the decimal represents tenths, the second decimal place represents hundredths, the third decimal place represents thousandths, etc. For example, the number 7.3 is equivalent to 7 3/10. The number 20.45 is equivalent to 20 45/100.
9. Not all fractions can be represented by terminating decimal numbers. For example, 1/3 has the decimal equivalent of 0.333333... a repeating, non-terminating decimal (it goes on forever).
10. Fractions are also a way of writing a division statement. 4/5 means 4 divided by 5. You can use this concept to find the decimal equivalent for a fraction: 4 divided by 5 = 0.8.
11. Percents represent parts out of 100. 50% means 50 parts out of 100. This is equivalent to the fraction 50/100, or the decimal 0.5.
Assignment: Use the first section of the Student Packet (Basic Fraction Concepts Assessment) to assess your students' understanding of basic fraction ideas. Since the handout is in Microsoft Word, you can add, delete or change questions as you see fit. Students who don't have a grasp of these basic ideas in 5th grade or above probably need additional support to learn them before going further.
Basic ideas about decimal numbers and percents (Decimals and Percents Assessment) are assessed in the last section of the Student Packet.